Exponential equation trouble?

nightshade52 · May 15, 2021

Hi. I am attempting to find/come up with an equation to express a particular pattern, but I am having difficulty figuring out how to get it to produce the numbers that I desire. Here is the gist of what I am trying to accomplish. I have a range of 1 to 500,000,000, and I want to increase a number by 1 for every step from 1 to 20, but only when the first number is doubled, and the first doubling of that number is 1907. In other words, from 1-1906, the second number would be from 1 to <2. At 1907 the second number is 2, then when 1907 is doubled, the second number is 3, etc... all the way until the first number is 500,000,000 and the second number is 20. I believe this is a form of exponential equation, but it's been a long time. Also, not entirely sure this is the right place for this question. Thank you in advance.

Dr.Peterson · May 15, 2021

nightshade52 said:
Hi. I am attempting to find/come up with an equation to express a particular pattern, but I am having difficulty figuring out how to get it to produce the numbers that I desire. Here is the gist of what I am trying to accomplish. I have a range of 1 to 500,000,000, and I want to increase a number by 1 for every step from 1 to 20, but only when the first number is doubled, and the first doubling of that number is 1907. In other words, from 1-1906, the second number would be from 1 to <2. At 1907 the second number is 2, then when 1907 is doubled, the second number is 3, etc... all the way until the first number is 500,000,000 and the second number is 20. I believe this is a form of exponential equation, but it's been a long time. Also, not entirely sure this is the right place for this question. Thank you in advance.

As I understand this, you want a function that takes as input a number from 1 to 500,000,000, so that when the input is 1, the output is 1, and it increases until for input 1907 it is 2, and thereafter it increases by 1 whenever the input doubles. So it would look like this:

input ... output
1 ... 1
1907 ... 2
3814 ... 3
...
500,000,000 ... 20

This appears to be mostly a logarithmic function (which is the inverse of an exponential function). Specifically, the output would be \(\log_2(\frac{x}{1907}) + 1\).

But ...

That would be 0 for input 956.5, and negative before that; so you want to force a minimum of 1.

And for input 500,000,000, the output would be 19, so either you or I seem to be off by 1.

Here is a table from Excel for the function I proposed:

x	log_2(x/1907)+1
1	-9.89709
1907	1.00000
3814	2.00000
5721	2.58496
7628	3.00000
9535	3.32193
11442	3.58496
13349	3.80735
15256	4.00000
17163	4.16993
19070	4.32193
20977	4.45943
22884	4.58496
24791	4.70044
26698	4.80735
28605	4.90689
30512	5.00000
32419	5.08746
34326	5.16993
36233	5.24793
38140	5.32193
40047	5.39232
41954	5.45943
43861	5.52356
45768	5.58496
47675	5.64386
49582	5.70044
51489	5.75489
53396	5.80735
55303	5.85798
57210	5.90689
59117	5.95420
61024	6.00000
500000000	19.00026

nightshade52 · May 15, 2021

Dr.Peterson said:
As I understand this, you want a function that takes as input a number from 1 to 500,000,000, so that when the input is 1, the output is 1, and it increases until for input 1907 it is 2, and thereafter it increases by 1 whenever the input doubles. So it would look like this:

input ... output
1 ... 1
1907 ... 2
3814 ... 3
...
500,000,000 ... 20

This appears to be mostly a logarithmic function (which is the inverse of an exponential function). Specifically, the output would be \(\log_2(\frac{x}{1907}) + 1\).

But ...

That would be 0 for input 956.5, and negative before that; so you want to force a minimum of 1.

And for input 500,000,000, the output would be 19, so either you or I seem to be off by 1.

Here is a table from Excel for the function I proposed:

x
log_2(x/1907)+1
1
-9.89709
1907
1.00000
3814
2.00000
5721
2.58496
7628
3.00000
9535
3.32193
11442
3.58496
13349
3.80735
15256
4.00000
17163
4.16993
19070
4.32193
20977
4.45943
22884
4.58496
24791
4.70044
26698
4.80735
28605
4.90689
30512
5.00000
32419
5.08746
34326
5.16993
36233
5.24793
38140
5.32193
40047
5.39232
41954
5.45943
43861
5.52356
45768
5.58496
47675
5.64386
49582
5.70044
51489
5.75489
53396
5.80735
55303
5.85798
57210
5.90689
59117
5.95420
61024
6.00000
500000000
19.00026

Thank you so much! This is what I was missing. I have completely forgotten how to do logs, to the point that I hadn't even considered it. Thank you. All I had to do to make the values all work is change it from +1 to +2.

nasi112 · May 16, 2021

Dr.Peterson said:
As I understand this, you want a function that takes as input a number from 1 to 500,000,000, so that when the input is 1, the output is 1, and it increases until for input 1907 it is 2, and thereafter it increases by 1 whenever the input doubles. So it would look like this:

input ... output
1 ... 1
1907 ... 2
3814 ... 3
...
500,000,000 ... 20

This appears to be mostly a logarithmic function (which is the inverse of an exponential function). Specifically, the output would be \(\log_2(\frac{x}{1907}) + 1\).

But ...

That would be 0 for input 956.5, and negative before that; so you want to force a minimum of 1.

And for input 500,000,000, the output would be 19, so either you or I seem to be off by 1.

Here is a table from Excel for the function I proposed:

x
log_2(x/1907)+1
1
-9.89709
1907
1.00000
3814
2.00000
5721
2.58496
7628
3.00000
9535
3.32193
11442
3.58496
13349
3.80735
15256
4.00000
17163
4.16993
19070
4.32193
20977
4.45943
22884
4.58496
24791
4.70044
26698
4.80735
28605
4.90689
30512
5.00000
32419
5.08746
34326
5.16993
36233
5.24793
38140
5.32193
40047
5.39232
41954
5.45943
43861
5.52356
45768
5.58496
47675
5.64386
49582
5.70044
51489
5.75489
53396
5.80735
55303
5.85798
57210
5.90689
59117
5.95420
61024
6.00000
500000000
19.00026

Dr.Peterson, you have a great brain inside your head

Dr.Peterson · May 16, 2021

nasi112 said:
Dr.Peterson, you have a great brain inside your head

Except when I add 1 instead of 2 and show a table of values that contradicts my own claim. That's what I get for not rechecking the first numbers ...

nasi112 · May 16, 2021

Dr.Peterson said:
Except when I add 1 instead of 2 and show a table of values that contradicts my own claim. That's what I get for not rechecking the first numbers ...

still great.
At least, the OP got what he/she needs

Exponential equation trouble?

nightshade52

New member

Dr.Peterson

Elite Member

nightshade52

New member

nasi112

Full Member

Dr.Peterson

Elite Member

nasi112

Full Member